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A111152 Smallest number that is a sum of two n-th powers of positive rationals but not of two n-th powers of positive integers. 2
6, 5906, 68101, 164634913, 69071941639 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,1

COMMENTS

a(6) and a(7) are only conjectures; the earlier terms have (apparently) been proved.

LINKS

Table of n, a(n) for n=3..7.

A. Bremner and P. Morton, A new characterization of the integer 5906, Manuscripta Math. 44 (1983) 187-229; Math. Rev. 84i:10016.

S. R. Finch, On a Generalized Fermat-Wiles Equation

Dave Rusin, Seeking counterexamples to FLT in other rings [Broken link]

Dave Rusin, Seeking counterexamples to FLT in other rings [Cached copy]

EXAMPLE

a(3) = 6 = (17/21)^3 + (37/21)^3

a(4) = 5906 = (25/17)^4 + (149/17)^4

a(5) = 68101 = (15/2)^5 + (17/2)^5

a(6) <= 164634913 = (44/5)^6 + (117/5)^6 (John W. Layman, Oct 20, 2005)

a(7) <= 69071941639 = (63/2)^7 + (65/2)^7

From a posting to the Number Theory Mailing List by Seiji Tomita (fermat(AT)M15.ALPHA-NET.NE.JP), Sep 10 2009: (Start)

a(8) <= (50429/17)^8 + (43975/17)^8

a(9) <= (257/2)^9 + (255/2)^9

a(10) <= (1199/5)^10 + (718/5)^10

a(11) <= (1025/2)^11 + (1023/2)^11

a(12) <= (9298423/17)^12 + (8189146/17)^12

a(13) <= (4097/2)^13 + (4095/2)^13

a(14) <= (76443/5)^14 + (16124/5)^14

a(15) <= (16385/2)^15 + (16383/2)^15

a(16) <= (3294416782861362/97)^16 + (2731979866522411/97)^16

a(17) <= (65537/2)^17 + (65535/2)^17

a(18) <= (1721764/5)^18 + (922077/5)^18

a(19) <= (262145/2)^19 + (262143/2)^19

a(20) <= (726388197629/17)^20 + (86503985645/17)^20

(End)

CROSSREFS

Sequence in context: A116135 A011788 A172649 * A202969 A003191 A298272

Adjacent sequences:  A111149 A111150 A111151 * A111153 A111154 A111155

KEYWORD

nonn,more,hard

AUTHOR

David W. Wilson, Oct 19 2005

STATUS

approved

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Last modified December 16 15:11 EST 2018. Contains 318172 sequences. (Running on oeis4.)